Vector-valued smoothing for finite Sidon sets

Abstract

Let $F(N)$ denote the largest cardinality of a Sidon subset of $\{0,1,\dots ,N-1\}$. We prove \[ F(N)\le N^{1/2}+0.9435N^{1/4}+O(1). \] The argument uses a vector-valued convolution inequality in which several smoothing kernels cooperate to produce a boundary majorant while their $L^2$ energies are averaged. For fixed kernels and mixing weights, the boundary optimization is a strictly convex quadratic program with an explicit dual. An eight-kernel numerical candidate is converted into a finite rational certificate and verified using exact arithmetic only.

10 pages. A finite optimization framework and numerical search results have been added, and the coefficient has been improved to 0.9435